[This is not a solution, or even a hint, but it might give you a push in the right direction.]
To get an idea of what is going on here, start by looking at the very simplest situation, when the function F is constant, say . Then , so . This is a rank-one operator (you may perhaps have seen the notation for such an operator).
Next, take a slightly more complicated situation, where for . (Admittedly, that function F is not continuous, but the idea is to use it as a step function giving a Riemann approximation to a continuous function.) Then . So , or . This is again a finite-rank operator.
Now use the Riemann approximation idea. Given a continuous function , let , and try to show that as the operators defined as in the previous paragraph converge in norm to T. Then T will be a norm limit of finite-rank operators, hence compact.